RAFAEL BRAVO DE LA PARRA
	"DESTOBIO 2000" August 23-27, 2000 West Lafayette, Indiana, USA

"Large Aggregation of Systems with Different Time Scales: Applications to Population Dynamics."

ABSTRACT: Modeling biological systems, particularly ecological ones, we usually find very complex systems that we should try to manage to get some insights. A first method to do this consists in building an abstract model describing the real system in detail. This leads to a family of models involving a very large number of variables. The complexity of the system is included in the model and computer simulation becomes the only available tool to carry out its study. At the other extreme we can find those models avoiding almost every detail in order to be mathematically tractable. These models of ecological communities only deal with a few variables, assuming that the internal structure of the population has no important effect and can be neglected. This assumption corresponds to an approximation of the total system by means of a reduced one that should be checked. However, in most cases, simplified models are used and few arguments are given to justify them.

A third approach is available for some complex systems whose dynamics includes two or more different time scales. The so-called aggregation methods describe general complex systems with different time scales which could be studied by means of a reduced system. The reduced system, or aggregated system, must reflect in a certain way the two different dynamics involved in the general one, the one corresponding to the fast time scale and the one corresponding to the slow time scale. The slow dynamics of the general system, the initial complex one, usually corresponds to the dynamics of the reduced system, meanwhile the fast dynamics of the general system is reflected in the coefficients of the reduced one in such a way that it is possible to study the influences between the different hierarchical levels, which seems meaningful from an ecological point of view.

We present the aggregation methods for different kinds of dynamical systems: continuous or discrete, finite or infinite dimensional, and deterministic or stochastic.

We present several cases of aggregation applied to population dynamics. For systems of ODE's we study the effects of individual migration decisions on the dynamics of populations in an heterogeneous environment. In another ODE's case we develop the coupling between individual behavior, described by game dynamics, and population dynamics at different time scales.

For discrete systems we develop models including a migration process and a demographic process at two different time scales. We will consider different cases with migration and demography density dependent or independent, and any one of the two processes fast in comparison to the other one.

The main conference page

The Department of Mathematics at Purdue

Purdue University Campus Map

hits since 4/14/00.